Addition and Subtraction with Scientific Notation

Adding and subtracting with scientific notation may require more care, because the rule for adding and subtracting exponential expressions is that the expressions must havelike terms. Remember that to be like terms, two expressions must have exactly the same base numbers to exactly the same powers. Thinking about decimal arithmetic, the requirement that we have the same powers makes sense, because that guarantees that all of the place values are lined up properly.

Example: (4.5 × 104) + (1.75 × 104) can be completed using the distributive property of multiplication over addition, i.e., factor out the common factor 104.

We run into trouble, though, with problems like (7.5 × 103) + (5.25 × 105) because the powers of 10 differ, so we need to modify the problem before we factor. We work around this by using our exponent property bm · bn = b(m+n) to rewrite the 105 as 102 · 103 and then grouping the 102 with the 5.25.

We see that this solution is not in standard scientific notation form because the decimal part has more than one digit in front of the decimal point. So we have one more step to finish the problem. We need to rewrite 532.5 as 5.325 × 102 and then simplify the powers of ten.

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Subtraction can be done the same way as addition, by getting the powers of ten to match; factoring out the power of ten that is the same, and subtracting the decimal values that come together when the power of ten is factored out. Then we simplify if the answer is not in normalized form.

Self-Check Problem

Approximately, how much further from the sun is Saturn than Earth. Earth is approximately 9.3 × 107 miles from the sun and Saturn is approximately 8.87 × 108 miles from the sun.